Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is compared. I (...) conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

Abstract Objections to life extension often focus on its effects for individual well-being. Prominent amongst these concerns is the possibility that life extending technologies will extend lifespan without preventing the ageing of the mind. Writers on the subject express the fear that life extending drugs will keep us physically youthful whilst our minds decay, succumbing to dementia, boredom, and loneliness. Generally these fears remain speculative, in part due to the absence of genuine life extending technologies. In this paper, however, I (...) examine the implications of an existing life extension technology. Caloric restriction (CR) and drugs that mimic its effects, such as rapamycin, metformin and resveratrol have been shown to increase average and maximum lifespan in a wide variety of organisms, and seem likely to do so in humans. Moreover, some CR mimetic drugs (CRMs) are already widely used. This means that they present a pressing test case for fears about mental ageing in an extended life. Misgivings about mental ageing can be divided into biomedical factors such as the likelihood of brain ageing, and psychological factors such as loss of meaning and boredom. I argue that studies of CR suggest that brain ageing will be beneficially slowed. However, it is less clear that deleterious aspects of psychological ageing can be similarly retarded. I argue that this reduces the desirability of life extension unless major social changes can be made. (shrink)

There is growing public interest in developing biomedical technologies capable of extending the human lifespan. This chapter discusses the desirability of a substantial life extension with regard to its implications for personal identity over time. The possibility of significant lifespan extension involves two different personal‐identity questions. First, it raises the question of whether one is identifiable as the very same person throughout a lifetime; secondly, it challenges our self‐conception. The chapter describes the way in which personal identity as “sameness” might (...) be affected by the prospect of substantial life extension. It then focuses on the significance of the conception of personal identity as “selfhood” for radically extended lifespans. Finally, the chapter argues that a substantial life extension raises important concerns about the continuity of one's identity over time and is not straightforwardly desirable for the individual. (shrink)

It seems to be a widespread opinion that increasing the length of existing happy lives is better than creating new happy lives and that it may be better even when the total welfare is lower in the outcome with extended lives. The chapter discusses two interesting suggestions that seem to support this idea. The first is critical level utilitarianism (CLU) and the other is view comparativism. The chapter describes the pure case of life extension versus life replacement and then presents (...) some different views about the value of life extension, indicating some of the arguments in favor and against life extension fail. Then, it turns to the implications of critical level utilitarianism and comparativism in regards to life extension versus replacement, the main topic of this chapter. A case is presented to explain that there is a conflict between intuitions regarding life extension and comparativism. (shrink)

Extended simple regions have been the focus of recent developments in philosophical logic, metaphysics, and philosophy of physics. However, only a handful of works provides a rigorous characterization of an extended simple region. In particular, a recent paper in this journal defends a definition based on an extrinsic notion of least distance. Call it the Least Distance proposal. This paper provides the first assessment of it. It argues that Least Distance faces difficulties and drawbacks. The paper then goes on to (...) suggest a different proposal, the Diameter proposal that is able to handle such drawbacks and difficulties. (shrink)

Extensive measurement is the standard measurement-theoretic approach for constructing a ratio scale. It involves the comparison of objects that can be concatenated in an additively representable way. This paper studies the implications of extensively measurable welfare for social choice theory. We do this in two frameworks: an Arrovian framework with a fixed population and no interpersonal comparisons, and a generalized framework with variable populations and full interpersonal comparability. In each framework we use extensive measurement to introduce novel domain restrictions, independence (...) conditions, and constraints on social evaluation. We prove a welfarism theorem for these domains and characterize the social welfare functions that satisfy the axioms of extensive measurement at both the individual and social levels. The main results are simple axiomatizations of strong dictatorship in the Arrovian framework and classical utilitarianism in the generalized framework. (shrink)

I discuss Clark’s predictive processing/extended mind hybrid, diagnosing a problem: Clark’s hybrid suggests that, when we use them, we pay attention to mind-extending external resources. This clashes with a commonly accepted necessary condition of cognitive extension; namely, that mind-extending resources must be phenomenally transparent when used. I then propose a solution to this problem claiming that the phenomenal transparency condition should be rejected. To do so, I put forth a parity argument to the effect that phenomenal transparency cannot be a (...) necessary condition on cognitive extension: roughly, since internal cognitive resources can fail to be transparent when used, by parity, external resources can fail to be phenomenally transparent too. Further, I argue that phenomenal transparency is not even a reliable indicator of cognitive extension; hence its absence should not be considered a problem for Clark’s extended mind-predictive processing hybrid. Lastly, I consider and allay a number of worries my proposal might raise, and conclude the paper. (shrink)

This study focuses on the devices implemented in classical Indian texts on ritual and language in order to develop a structure of rules in an economic and systematic way. These devices presuppose a spatial approach to ritual and language, one which deals for instance with absences as substitutions within a pre-existing grid, and not as temporal disappearances. In this way, the study reveals a key feature of some among the most influential schools of Indian thought. The sources are Kalpasūtra, Vyākaraṇa (...) and Mīmāṃsā, three textual traditions which developed alongside each other, sharing - as the volume shows - common presuppositions and methodologies. The book will be of interest for Sanskritists, scholars of ritual exegesis and of the history of linguistics. - Cover. (shrink)

We analyze extensive games as interactive process models, using modallanguages plus matching notions of bisimulation as varieties of gameequivalences. Our technical results show how to fit existing modalnotions into this new setting.

In recent years, many people writing on set theory have invoked the notion of an indefinitely extensible concept. The notion, it is usually claimed, plays an important role in solving the paradoxes of absolute infinity. It is not clear, however, how the notion should be formulated in a coherent way, since it appears to run into a number of problems concerning, for example, unrestricted quantification. In fact, the notion makes perfectly good sense if one endorses a dialetheic solution to the (...) paradoxes. It then morphs from a supposed solution to the paradoxes into a diagnosis of their structure. In this paper I show how. (shrink)

This paper investigates the role of a pre-existing body-model that is an enabling constraint for the incorporation of objects into the body. This body-model is also a basis for the distinction between body extensions (e.g., in the case of tool-use) and incorporation (e.g., in the case of successful prosthesis use). It is argued that, in the case of incorporation, changes in the sense of body-ownership involve a reorganization of the body-model, whereas extension of the body with tools does not involve (...) changes in the sense of body-ownership. (shrink)

"La Nouvelle Alliance (1979), by physicist Ilya Prigogine and philosopher Isabelle Stengers, is both a book on the history and philosophy of thermodynamics and a philosophical essay on man's place in nature. It has given rise to lively interdisciplinary controversy.".

When we remember a scene, the scene’s boundaries are wider than the boundaries of the scene we saw. This phenomenon is called boundary extension. The most important philosophical question about boundary extension is whether it is a form of perceptual adjustment or adjustment during memory encoding. The aim of this paper is to propose a third explanatory scheme, according to which the extended boundary of the original scene is represented by means of mental imagery. And given the similarities between perception (...) and mental imagery, the memory system encodes both the part of the scene that is represented perceptually and the part of the scene that is represented by means of mental imagery. (shrink)

This article introduces and motivates the notion of a “properly extensive” quantity by means of a puzzle about the reliability of certain canonical length measurements. An account of these measurements’ success, I argue, requires a modally robust connection between quantitative structure and mereology that is not mediated by the dynamics and is stronger than the constraints imposed by “mere additivity.” I outline what it means to say that length is not just extensive but properly so and then briefly sketch an (...) application of proper extensiveness to the project of providing a reductive ground for metric quantitative structure. (shrink)

On the ethics of extending human life: healthy people have a right to carry on livingHumanity has long demonstrated a paradoxical ambivalence concerning the extension of a healthy human lifespan. Modest health extension has been universally sought, whereas extreme health extension has been regarded as a snare and delusion—a dream beyond all others at first blush, but actually something we are better off without. The prevailing pace of biotechnological progress is bringing ever closer the day when humanity will be able (...) to act on the latter view by rejecting a clear and present opportunity for much longer healthy lives. Indeed, some biogerontologists contend that that day has already arrived, to the extent that our hesitation in embarking on a vigorous “war on ageing” is already delaying the point at which a cure for ageing will be developed. Here I consider whether our present caution concerning the wisdom of truly curing ageing is likely to survive the increased scrutiny that it will receive in coming years as a result of these technical advances. I conclude that it will not, because of its irreconcilability with values that are more deeply held by the large majority of humanity than any values that argue against the quest for a cure. I further conclude that all the major current reasons given for not curing ageing are mere crutches to help us cope with the immutability of ageing that we have been brought up to accept. Our failure to set aside such irrationality is already shortening potential longevity—quite probably of those already alive today—to a staggering degree. Once we realise this, our determination to consign human ageing to history will be second …. (shrink)

Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from (...) MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses. (shrink)

Radical enactive and embodied approaches to cognitive science oppose the received view in the sciences of the mind in denying that cognition fundamentally involves contentful mental representation. This paper argues that the fate of representationalism in cognitive science matters significantly to how best to understand the extent of cognition. It seeks to establish that any move away from representationalism toward pure, empirical functionalism fails to provide a substantive “mark of the cognitive” and is bereft of other adequate means for individuating (...) cognitive activity. It also argues that giving proper attention to the way the folk use their psychological concepts requires questioning the legitimacy of commonsense functionalism. In place of extended functionalism—empirical or commonsensical—we promote the fortunes of extensive enactivism, clarifying in which ways it is distinct from notions of extended mind and distributed cognition. (shrink)

Any inquiry into the meaning and implications of the prolongation of the human lifespan requires an investigation of its metaphysical basis and its ethical outcomes. This chapter explains a series of metaphysical and ethical claims about lifespan extension. It highlights a number of arguments that are typically put forward against these claims, and shows the ways in which they are mistaken. Two such claims given in the chapter are: (1) aging and life stages are neither wholly constituted by biological givens, (...) nor wholly understandable in terms of biological parameters; instead, aging and life stages are, in crucial ways, socially constructed; and (2) death is bad, and other things being equal, a longer life is a better life. (shrink)

According to an influential idea in the philosophy of set theory, certain mathematical concepts, such as the notion of a well-order and set, are indefinitely extensible. Following Parsons (1983), this has often been cashed out in modal terms. This paper explores instead an extensional articulation of the idea, formulated in higher-order logic, that flat-footedly formalizes some remarks of Zermelo. The resulting picture is incompatible with the idea that the entire universe can be well-ordered, but entirely consistent with the idea that (...) the sets of any set-theoretic universe can be. (shrink)

In this chapter I provide an elementary exposition of the development of Whitehead's view of objectivity and his theory of extension, with particular focus on explaining how macroscopic objects of ordinary perception and the whole structure of space-time arise out of the units of his ontology, namely, the actual occasions. I also review the scholarship on Whitehead's view of extension and discuss the major problems that arise in connection with the theory in his magnum opus, Process and Reality.

We give alternative proofs of results due to Paris and Wilkie concerning the existence of end extensions of countable models of $B\Sigma_{1}$, that is, the theory of $\Sigma_{1}$ collection.

Arguments from fairness as described in Aristotle’s _Rhetoric_ are usually taken to aim at mitigating the strictness of the law or, in terms of procedure, to favour the defendant. This paper considers a more inclusive interpretation, that is, that arguments from fairness can work both ways. In the example given in the _Rhetoric,_ arguments from fairness are directed at a restrictive interpretation of the text. That may not be necessary however. Likewise, fairness may speak for the claimant. Two examples may (...) support this conclusion: a judicial speech by Hyperides, and the doctrine of issues appearing in Hellenistic school rhetoric. (shrink)

During the past decade, the so-called “hypothesis of cognitive extension,” according to which the material vehicles of some cognitive processes are spatially distributed over the brain and the extracranial parts of the body and the world, has received lots of attention, both favourable and unfavourable. The debate has largely focussed on three related issues: (1) the role of parity considerations, (2) the role of functionalism, and (3) the importance of a mark of the cognitive. This paper critically assesses these issues (...) and their interconnections. Section 1 provides a brief introduction. Section 2 argues that some of the most prominent objections against the appeal to parity considerations fail. Section 3 shows that such considerations are nevertheless unsuitable as an argument for cognitive extension. First, the actual argumentative burden is carried by an underlying commitment to functionalism, not by the parity considerations themselves. Second, in the absence of an independently motivated mark of the cognitive, the argument based on parity considerations does not get off the ground, but given such a mark, it is superfluous. Section 4 argues that a similar dilemma arises for the attempt to defend cognitive extension by a general appeal to functionalism. Unless it can be independently settled what it is for a process to be cognitive, functionalism itself will be undermined by the possibility of cognitive extension. Like parity considerations, functionalism is thus either unable to support cognitive extension or superfluous. Hence, nothing short of the specification of an appropriate mark of the cognitive that can be fulfilled not only by intracranial but also by extended processes will do as an argument for cognitive extension. (shrink)

The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for iterated contours of sequential cascades. A related problem of continuous extension of maps between maximal elements of sequential cascades to full subcascades is solved in full generality.

Dummett's account of the semantic paradoxes in terms of his theory of indefinitely extensible concepts is compared with Bürge's account in terms of indexicality. Dummett's appeal to intuitionistic logic does not block the paradoxes but Bürge's attempt to avoid the Strengthened Liar is unconvincing. It is argued that in order to avoid the Strengthened Liar and other semantic paradoxes involving nonindexical expressions (constants), one must postulate that when we reflect on the paradoxes there are slight shifts in the meaning (not (...) just reference) we ascribe to metalinguistic expressions (in particular 'say', and derivatively 'true' and 'false'). Consideration of metaphor and gradual linguistic change suggests that such semantic shifts are consistent with language-learning and communication. On this account there is no threat to classical logic, bivalence or the fundamental principles governing 'true' and 'false'. (shrink)

This paper brings together several strands of thought from both the analytic and phenomenological traditions in order to critically examine accounts of cognitive enhancement that rely on the idea of cognitive extension. First, I explain the idea of cognitive extension, the metaphysics of mind on which it depends, and how it has figured in recent discussions of cognitive enhancement. Then, I develop ideas from Husserl that emphasize the agential character of thought and the distinctive way that conscious thoughts are related (...) to one another. I argue that these considerations are necessary for understanding why forms of cognitive extension may diminish our cognitive lives in different ways. This does not lead to a categorical rejection of cognitive enhancement as unethical or bad for human flourishing, but does warrant a conservative approach to the design and implementation of cognitive artifacts. (shrink)

Public extension services play a key role in the implementation of strategies for rural development based on the sustainable management of natural resources. However, the sector suffers from restricted financial and human resources. Using experiences from participatory action research, a strategy for rural extension in the Amazon was defined to increase the efficiency and the relevance of external support for local resource users. This strategy considered activities initiated and coordinated by local people. Short-term facilitation visits provided continuous external support for (...) the purpose of establishing locally based planning and learning mechanisms. In collaboration with the municipality of Muaná in the Eastern Amazon, the strategy was tested in two traditional communities – Monte Moriat and Boa Esperança. Both communities recognized as most important the need to reduce their dependence on açaí (Euterpe oleracea Mart.), the principal source of food and income. This forest resource has been overexploited, seriously affecting açaí stands and diminishing the forest benefits fundamental for survival. Two local groups decided to raise chickens as a way to reduce nutritional problems and as an alternative source of income. Supported by monthly, short-term planning and evaluation visits, the groups managed to achieve significant progress in their endeavor. The groups identified (1) the control of decision-making and information, and (2) the absence of external obligations to be the greatest advantages of the development initiative. The study confirmed the potential of participatory strategies for public extension in the Amazon. Special attention was given to realistically define the role of forests for local development. However, drastic measures are necessary to smooth the way for real participation in governmental and non-governmental organizations acting in the Brazilian Amazon. (shrink)

The object of this paper is to examine half and full connexive extensions of the basic regular conditional logic CR. Extensions of this system are of interest because it is among the strongest well-known systems of conditional logic that can be augmented with connexive theses without inconsistency resulting. These connexive extensions are characterized axiomatically and their relations to one another are examined proof-theoretically. Subsequently, algebraic semantics are given and soundness, completeness, and decidability are proved for each system. The semantics is (...) also used to establish independence results. Finally, a deontic interpretation of one of the systems is examined and defended. (shrink)

In a recent publication in this journal, Asle Kiran and Peter-Paul Verbeek (hereafter K&V) argue that extension theory and the notion of trust it implies are flawed. In this commentary, I defend extension theory against their critique. I first briefly introduce extension theory, then reconstruct K&V’s five arguments against extension theory and demonstrate that four of their five arguments are misplaced.

The principle of sufficient reason threatens modal collapse. Some have suggested that by appealing to the indefinite extensibility of contingent truth, the threat is neutralized. This paper argues that this is not so. If the indefinite extensibility of contingent truth is developed in an analogous fashion to the most promising models of the indefinite extensibility of the concept set, plausible principles permit the derivation of modal collapse.

Intuitionistic sentential logic is shown to be sound and complete with respect to a semantics centered around extensions of atomic bases (i.e. sets of inference rules for atomic sentences). The result is made possible through a non-standard interpretation of disjunction, whereby, roughly speaking, a disjunction is taken to hold just in case every atomic sentence that follows from each of the disjuncts separately holds; it is argued that this interpretation makes good sense provided that rules in atomic bases are conceived (...) of as being accepted hypothetically rather than categorically. (shrink)

Extension is probably the most general natural property. Is it a fundamental property? Leibniz claimed the answer was no, and that the structureless intuition of extension concealed more fundamental properties and relations. This paper follows Leibniz's program through Herbart and Riemann to Grassmann and uses Grassmann's algebra of points to build up levels of extensions algebraically. Finally, the connection between extension and measurement is considered.

In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(Λ). This summarizes results that can be found already in [13, 14] and [4]. Furthermore, we determine the (...) structure of the lattice of extensions of n(LC). (shrink)

Scientists, bioethicists, and policy makers are currently engaged in a contentious debate about the scientific prospects and morality of efforts to increase human longevity. Some demographers and geneticists suggest that there is little reason to think that it will be possible to significantly extend the human lifespan. Other biodemographers and geneticists argue that there might well be increases in both life expectancy and lifespan. Bioethicists and policy makers are currently addressing many of the ethical, social, and economic issues raised by (...) life extension research. However, the emphasis on philosophical argument supporting or condemning efforts to increase human longevity means that much less attention is currently being given to the factors that might play a role in generating interest in efforts to increase human longevity. This analysis considers three factors that might play a role in heightening public interest in efforts to develop biomedical technologies capable of retarding or reversing aging processes. While discussions of life extension research can seem quite futuristic and impractical, there are some powerful existential factors that might well generate considerable public support for life extension strategies if effective biomedical interventions emerge. Rather than providing philosophical justifications supporting or condemning efforts to increase human longevity, this essay seeks to promote a better understanding of the factors generating contemporary interest in prolonging life and postponing death. (shrink)

Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. This is interpreted as involving (...) arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement. (shrink)

When we remember a scene, the scene’s boundaries are wider than the boundaries of the scene we saw. This phenomenon is called boundary extension. The most important philosophical question about boundary extension is whether it is a form of perceptual adjustment or adjustment during memory encoding. The aim of this paper is to propose a third explanatory scheme, according to which the extended boundary of the original scene is represented by means of mental imagery. And given the similarities between perception (...) and mental imagery, the memory system encodes both the part of the scene that is represented perceptually and the part of the scene that is represented by means of mental imagery. (shrink)

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

Few philosophical issues have had as long and elaborate a history as the problem of free will, which has been contested at every stage of the history of the subject. The present work practices an extensive bibliography of this elaborate literature, listing some five thousand items ranging from classical antiquity to the present.

In this paper, we look at applying the techniques from analyzing superintuitionistic logics to extensions of the cointuitionistic Priest-da Costa logic daC (introduced by Graham Priest as “da Costa logic”). The relationship between the superintuitionistic axioms- definable in daC- and extensions of Priest-da Costa logic (sdc-logics) is analyzed and applied to exploring the gap between the maximal si-logic SmL and classical logic in the class of sdc-logics. A sequence of strengthenings of Priest-da Costa logic is examined and employed to pinpoint (...) the maximal non-classical extension of both daC and Heyting-Brouwer logic HB . Finally, the relationship between daC and Logics of Formal Inconsistency is examined. (shrink)

According to Reid, color sensations are not extended nor are they arranged in figured patterns. Reid further claimed that ‘there is no sensation appropriated to visible figure.’ Reid justified these controversial claims by appeal to Cheselden's report of the experiences of a young man affected by severe cataracts, and by appeal to cases of perception of visible figure without color. While holding fast to the principle that sensations are not extended, Dugald Stewart tried to show that ‘a variety of colour (...) sensations is a necessary means for the perception of visible figure.’ According to John Fearn, two motives appear to be central to Reid's views about color sensations and extension: his commitment to the Cartesian doctrine of the immateriality of the soul, and his attempt to evade ‘Hume's dilemma’ about the existence and immateriality of the soul. (shrink)

We show that every model of IΔ0 has an end extension to a model of a theory where log-space computable function are formalizable. We also show the existence of an isomorphism between models of IΔ0 and models of linear arithmetic LA.

This article presents an approach to the semantics of non-distributive propositional logics that is based on a lattice representation theorem that delivers a canonical extension of the lattice. Our approach supports both a plain Kripke-style semantics and, by restriction, a general frame semantics. Unlike the framework of generalized Kripke frames, the semantic approach presented in this article is suitable for modeling applied logics, as it respects the intended interpretation of the logical operators. This is made possible by restricting admissible interpretations.

. We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is , I is and O is , where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether is involutive (...) or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of by the logical operations , under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures. (shrink)

We introduce the notion of τ-like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω-like means that every element has finitely many predecessors, while being ζ-like means that every interval is finite. We consider statements of the form “any τ-like partial order has a τ-like linear extension” and “any τ-like partial order is embeddable into τ” . Working in the framework of reverse mathematics, we show that these (...) statements are equivalent either to equation image or to equation image over the usual base system equation image. (shrink)

The paper considers certain extensions of the system LC introduced in Dunn & Meyer 1997. LC is a structurally free system , but it has combinators as formulas in the place of structural rules. We consider two ways to extend LC with conjunction and disjunction depending on whether they distribute over each other or not. We prove the elimination theorem for the systems. At the end of the paper we give a Routley-Meyer style semantics for the distributive extension, including some (...) new definitions and an algorithm which are used in proving a representation theorem in general, i.e., for arbitrary combinators. (shrink)

Following considerable debate, the practice of euthanasia was legalized in Belgium in 2002, thereby making Belgium one of the few places in the world where this practice is legal. In 2014 the law was amended for the first time. The 2014 amendment makes euthanasia legally possible for all minors who repeatedly and voluntarily request euthanasia and who are judged to possess “capacity of discernment”, as well as fulfil a number of other criteria of due care. This extension of the 2002 (...) euthanasia law generated a lot of national and international debate and has been applauded by many and heavily criticized by others. This evolution is clearly of interest to end-of-life debates in the entire world. This paper will therefore describe how this amendment came to get passed using official documents from Belgium's Senate and Chamber of Representatives where this amendment was discussed and subsequently passed. Next, some of the most commonly given arguments in favour of the law are identified, as well as the arguments most often voiced against the amendment. All these arguments will be expanded upon and it will be examined whether they hold up to ethical scrutiny. Analysing the official documents and identifying the most commonly voiced arguments gives valuable insight into how Belgium came to amend its euthanasia law and why it did so in 2014. It also becomes clear that although the current amendment is often seen as far-reaching, more radical ideas were proposed during the drafting of the law. Also, in analysing those arguments in favour of the amendment and those against, it is clear that the validity of some of these is questionable. (shrink)

El artículo se propone caracterizar la Extensión Crítica, como una alternativa modalidad emergente, desde una perspectiva epistemológica. Caracteriza al conocimiento propio de la misma como orientado por un interés emancipador, lo que supone procesos de autorreflexión, concientización, crítica ideológica y praxis. Es un conocimiento práctico para la acción y que se construye en y desde la acción. Su modo de producción propone una epistemología que reconoce al otro como sujeto cognoscente y legitima reflexivamente sus saberes silenciados, proponiendo incluir el (...) punto de visa de los excluidos y, por lo tanto, una epistemología extendida, que articula conocimientos experienciales, proposicionales, representacionales y prácticos, así como científicos, populares, cotidianos y nativos. La complejidad de los problemas socio-políticos abordados desde la Extensión Crítica interpela a abordajes interdisciplinares y a la conformación de comunidades de prácticas. Finalmente, se proponen otros criterios de validación de este tipo de conocimiento, diferentes a los considerados en el campo de la extensión clásica: validez comunicativa, operatividad pragmática, empoderamiento, congruencia entre diferentes tipos de conocimiento y credibilidad. (shrink)